PhD Oral Exam - Akram Saleh, Electrical and Computer Engineering

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

Quantum error-correcting codes (QECCs) play a central role in the development of reliable quantum communication and quantum computation systems. Among the various classes of quantum codes, stabilizer codes are particularly important, as they serve as the quantum analogue of linear codes in classical error correction and can be constructed from suitable families of classical codes. A fundamental challenge in this area is to identify the conditions under which classical error-correcting codes can be employed to construct effective quantum codes.

The primary objective of this thesis is to establish necessary and sufficient conditions under which certain classes of classical codes give rise to high-performance quantum error-correcting codes. To this end, we focus on quasi-cyclic (QC) codes and their generalization, quasi-twisted (QT) codes, which possess rich algebraic structures and attractive implementation properties.

We first investigate construction algorithms for specific families of quasi-cyclic codes defined over finite commutative chain rings. By employing the Generalized Discrete Fourier Transform (GDFT), we develop an efficient and systematic algorithm for constructing generator matrices of repeated-root quasi-cyclic codes under particular constraints on the code length and the underlying ring. Unlike existing approaches that rely on exhaustive enumeration of constituent subcodes, the proposed method operates directly on generator matrices, resulting in substantial improvements in computational efficiency. Building on this framework, we further specialize the approach using the Discrete Fourier Transform to derive an algorithm for constructing generator matrices of a class of simple-root quasi-cyclic codes over finite chain rings.

To assess the practical impact of the proposed algorithms, we present a detailed performance comparison with existing construction methods. The evaluation focuses on encoding time per iteration and memory usage. Numerical results, illustrated using histograms, show that the proposed techniques achieve significantly faster encoding and reduced memory usage, with performance gains becoming more pronounced as the code length increases. We also list some classes of optimal and near optimal QC codes found through these proposed methods and some self-dual and self-orthogonal QC codes.

In addition, we analyze the algebraic structure of a class of repeated-root QC codes defined over finite fields. By interpreting these codes as linear codes over an appropriate auxiliary ring and introducing a suitable ring isomorphism, we establish a one-to-one correspondence between repeated-root QC codes over finite fields and a class of simple-root QT codes over finite chain rings. This correspondence allows known results for simple-root quasi-twisted codes over rings to be extended naturally to the repeated-root QC setting and leads to a clearer and more transparent structural characterization.

Building on the structural properties of these classical QC and QT codes, we study QT codes in terms of generator theory, especially one-generator QT codes of index three over finite fields. We characterize their dual codes with respect to the Hermitian inner product and derive sufficient conditions for self-orthogonality. Using these results, we construct examples of quantum error-correcting codes with good and, in some cases, optimal parameters in terms of minimum distance. Furthermore, we introduce a novel extension technique that preserves Hermitian self-orthogonality of this class of QT codes, enabling the construction of stabilizer quantum codes. Computational results include the discovery of binary stabilizer codes with optimal parameters.

Finally, motivated by the inherent self-orthogonality constraints in stabilizer code constructions, we explore entanglement-assisted quantum error-correcting codes (EAQECCs), which allow non-self-orthogonal classical codes to be utilized through shared entanglement between the sender and receiver. Focusing on the family of QT codes studied in this work, we propose construction techniques for maximal-entanglement EAQECCs. As a result, we obtain several new examples of maximal-entanglement EAQECCs with strong performance characteristics.